The present invention relates generally to three-dimensional (3D) computerized tomography (CT) and, more particularly, to methods and apparatus for reconstructing a 3D object image from incomplete x-ray cone beam projection data.
In conventional computerized tomography for both medical and industrial application, an x-ray fan beam and a linear array detector are employed. Two-dimensional (2D) imaging is achieved. While the data set is complete and image quality is correspondingly high, only a single slice of an object is imaged at a time. When a 3D image is required, a "stack Of slices" approach is employed Acquiring a 3D data set a 2D slice at a time is inherently tedious and time-consuming. Moreover, in medical applications, motion artifacts occur because adjacent slices are not imaged Simultaneously. Also, dose utilization is less than optimal, because the distance between slices is typically less than the x-ray collimator aperture, resulting in double exposure to many parts of the body.
A more recent approach, based on what is called cone beam geometry, employs a two-dimensional array detector instead of a linear array detector, and a cone beam x-ray source instead of a fan beam X-ray source. At any instant the entire object is irradiated by a cone beam x-ray source, and therefore cone beam scanning is much faster than slice-by-slice scanning using a fan beam or a parallel beam. Also, since each "point" in the object is viewed by the x-rays in 3D rather than in 2D much higher contrast can be achieved than is possible with conventional 2D x-ray CT. To acquire cone beam projection data, an object is scanned, preferably over a 360.degree. angular range, either by moving the x-ray source in an appropriate scanning trajectory, for example, a circular trajectory around the object, while keeping the 2D array detector fixed with reference to the source, or by rotating the object while the source and detector remain stationary. In either case, it is relative movement between the source and object which effects scanning.
However, image reconstruction procedures in x-ray CT are based on the Radon inversion process, in which the image of an object is reconstructed from the totality of the Radon transform of the object. The Radon transform of a 2D object consists of integrals of the object density on lines intersecting the object. The Radon transform of a 3D object consists of planar integrals. Image reconstruction by inversion from cone beam scanning data generally comprises two steps: (1) convert the cone beam data to planar integrals in Radon space, and (2) perform an inverse Radon transform on the planar integrals to obtain the image.
The cone beam geometry for 3D imaging has been discussed extensively in the literature, as represented by the following: Gerald N. Minerbo, "Convolutional Reconstruction from Cone-Beam Projection Data," IEEE Trans. Nucl. Sci., Vol. NS-26, No. 2, pp. 2682-2684 (Apr. 1979); Heang K. Tuy, "An Inversion Formula for Cone-Beam Reconstruction," SIAM J. Math., vol. 43, No. 3, pp. 546-552 (Jun. 1983) and Bruce D. Smith, "Image Reconstruction from Cone-Beam Projections: Necessary and Sufficient Conditions and Reconstruction Methods," IEEE Trans. Med. Imag., Vol. MI-44, pp. 1425 (Mar. 1985).
Depending on the scanning configuration employed to obtain the cone beam projection data, the data set in Radon space may be incomplete. While image reconstruction through inverse Radon transformation certainly can proceed, artifacts may be introduced, resulting in images which can be inadequate for medical diagnosis or part quality determination purposes.
A typical scanning and data acquisition configuration employing cone-beam geometry is depicted in FIG. 1. An object 20 is positioned within a field of view between a cone beam x-ray point source 22 and a 2D detector array 24, which provides cone beam projection data. An axis of rotation 26 passes through the field of view and object 20. For purposes of analysis, a midplane 28 is defined which contains the x-ray point source 22 and is perpendicular to the axis of rotation 26. By convention, the axis of rotation 26 is referred to as the z-axis, and the intersection of the axis of rotation 26 and the midplane 28 is taken as the origin of coordinates. x and y axes lie in the midplane 28 as indicated, and the (x,y,z) coordinate system rotates with the source 22 and detector 24. For scanning the object 20 at a plurality of angular positions, the source 22 moves relative to the object 20 and the field of view along a circular scanning trajectory 30 lying in the midplane 28, while the detector 24 remains fixed with respect to the source 22.
Thus, in the configuration of FIG. 1, data are acquired at a number of angular positions around the object by scanning the source and detector along the single circular scanning trajectory 30 (or equivalently rotating the object while the source and detector remain stationary). However, as demonstrated in the literature (e.g. Smith, 1985, above), and as described in greater detail hereinbelow, the data set collected in such a single scan is incomplete. As noted above, missing data in Radon space introduces artifacts during image reconstruction, resulting in images which can be inadequate for medical diagnosis or part quality determination purposes.
Smith (1985, above) has shown that a cone beam data set is complete if there is a point from the x-ray source scanning trajectory on each plane passing through the object of interest (with the assumptions that the detector is locked in position relative to the source and large enough to span the object under inspection). A configuration suggested by Minerbo (1979, above) and Tuy (1983, above), which Smith points out satisfies his condition for data completeness, is to employ two circular source scanning trajectories which are perpendicular to each other. Such a scanning configuration is however not always practical, as in the case of objects being very long in one dimension, such as a human body. Also, scanning in two perpendicular circles doubles the x-ray dosage to the object, which in some cases cannot be tolerated.
It may be noted that another scanning configuration which achieves data completeness is disclosed in commonly-assigned U.S. patent application Ser. No. 07/572,651, Filed Aug. 27, 1990, by Eberhard et al., and entitled "SQUARE WAVE CONE BEAM SCANNING TRAJECTORY FOR DATA COMPLETENESS IN THREE-DIMENSIONAL COMPUTERIZED TOMOGRAPHY." A scanning configuration which minimizes data incompleteness is disclosed in commonly-assigned U.S. patent application Ser. No. 07/572,590, filed Aug. 27, 1990, by Eberhard, and entitled "DUAL PARALLEL CONE BEAM CIRCULAR SCANNING TRAJECTORIES FOR REDUCED DATA INCOMPLETENESS IN THREE-DIMENSIONAL COMPUTERIZED TOMOGRAPHY." While effective to eliminate or reduce data set incompleteness, each of these approaches adds some complexity to the cone beam x-ray scanning configuration, for example by requiring motion in addition to rotation about the rotation axis, or by requiring additional x-ray sources and detectors. Additionally they increase the x-ray dose. Accordingly, the scanning geometry most commonly adopted is the circular scanning geometry depicted in FIG. 1.
In the context of the two general steps as stated above for image reconstruction by inversion from cone beam scanning data, it is relevant to note that the above-incorporated application Ser. No. 631,815 discloses efficient methods and apparatus for converting x-ray cone beam data to planar integrals, or values representing planar integrals, on a set of coaxial vertical planes in Radon space. The above-incorporated application Ser. No. 631,818 discloses a two-step approach for performing an inverse Radon transform starting with the planar integrals on the set of coaxial vertical planes. As Step 1 in the inverse Radon transform procedure, a 2D CT reconstruction procedure, such as filtered backprojection, is employed to calculate from the planar integrals a 2D projection image of the object on each of the planes. As Step 2, slices are defined in horizontal planes and the 3D image of the object is reconstructed slice-by-slice by employing for each slice a 2D CT reconstruction procedure, such as filtered backprojection, operating on the values of the 2D projection images in the plane of the slice to calculate a 2D image of the object for each slice.